Orthogonal polynomials in the spherical ensemble with two insertions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Byun, Sung-Soo Forrester, Peter J. Kuijlaars, Arno B. J. Lahiry, Sampad |
| author_facet | Byun, Sung-Soo Forrester, Peter J. Kuijlaars, Arno B. J. Lahiry, Sampad |
| contents | We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n, N\rightarrow\infty$ and $N-n$ fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted $L^2$ norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated $2\times 2$ Riemann-Hilbert problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15732 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orthogonal polynomials in the spherical ensemble with two insertions Byun, Sung-Soo Forrester, Peter J. Kuijlaars, Arno B. J. Lahiry, Sampad Classical Analysis and ODEs Mathematical Physics 42C05, 30C10, 30E25, 60B20, 41A60 We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n, N\rightarrow\infty$ and $N-n$ fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted $L^2$ norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated $2\times 2$ Riemann-Hilbert problem. |
| title | Orthogonal polynomials in the spherical ensemble with two insertions |
| topic | Classical Analysis and ODEs Mathematical Physics 42C05, 30C10, 30E25, 60B20, 41A60 |
| url | https://arxiv.org/abs/2503.15732 |